論文

© 2021, Springer-Verlag GmbH Germany, part of Springer Nature. The covariance matrix, which should be estimated from the data, plays an important role in many multivariate procedures, and its positive definiteness (PDness) is essential for the validity of the procedures. Recently, many regularized estimators have been proposed and shown to be consistent in estimating the true matrix and its support under various structural assumptions on the true covariance matrix. However, they are often not PD. In this paper, we propose a simple modification to make a regularized covariance matrix be PD while preserving its support and the convergence rate. We focus on the matrix ℓ∞ norm error in covariance matrix estimation because it could allow us to bound the error in the downstream multivariate procedure relying on it. Our proposal in this paper is an extension of the fixed support positive-definite (FSPD) modification by Choi et al. (2019) from spectral and Frobenius norms to the matrix ℓ∞ norm. Like the original FSPD, we consider a convex combination between the initial estimator (the regularized covariance matrix without PDness) and a given form of the diagonal matrix minimize the ℓ∞ distance between the initial estimator and the convex combination, and find a closed-form expression for the modification. We apply the procedure to the minimum variance portfolio (MVP) optimization problem and show that the vector ℓ∞ error in the estimation of the optimal portfolio weight is bounded by the matrix ℓ∞ error of the plug-in covariance matrix estimator. We illustrate the MVP results with S&P 500 daily returns data from January 1978 to December 2014.

While K-means is known to be a standard clustering algorithm, its performance may be compromised due to the presence of outliers and high-dimensional noisy variables. This paper proposes adaptively robust and sparse K-means clustering (ARSK) to address these practical limitations of the standard K-means algorithm. For robustness, we introduce a redundant error component for each observation, and this additional parameter is penalized using a group sparse penalty. To accommodate the impact of high-dimensional noisy variables, the objective function is modified by incorporating weights and implementing a penalty to control the sparsity of the weight vector. The tuning parameters to control the robustness and sparsity are selected by Gap statistics. Through simulation experiments and real data analysis, we demonstrate the proposed method’s superiority to existing algorithms in identifying clusters without outliers and informative variables simultaneously.